Journal of Convex Analysis Volume 3 (1996), No. 2, 167–194

A Characterization of Sets of Functions and Distributions on IR^{n} Described by Constraints on the Gradient

Antonio Corbo Esposito

Dipartimento di Ingegneria Industriale, Facoltà di Ingegneria, Università di Cassino, via Di Biasio 43, 03043 Cassino, Italy. e-mail: corbo@ing.unicas.it

Riccardo De Arcangelis

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, via Salvador Allende, 84081 Baronissi, Italy. e-mail: dearcang@matna2.dma.unina.it

Received October 18, 1994 Revised manuscript received March 14, 1996

Let U be a Hausdorff locally convex topological vector subspace of D′(IR^{n}) verifying suitable structure conditions. A characterization of the sets K ⊆ U that can be described as K = {u ∈ U: −⟨u, Dφ⟩ ∈ C for every φ ∈ D(IR^{n}) with φ ≥ 0, ∫_{IR_{n}} φ(x)dx = 1} for some closed convex subset C of IR^{n} is proved. As

corollaries characterizations of the sets K that can be described as K = {u ∈ Wloc^{1,p}(IR^{n}): Du ∈ C for a.e.

x in IR^{n}} or K = {u ∈ BVloc(IR^{n}): meas(A)−1 ∫_{A} dDu ∈ C for every nonempty bounded open set A of IR^{n}} for some closed convex subset C of IR^{n} are obtained. Similar results for subsets K of D′(IR^{n}), S′, Lloc^{p}(IR^{n}), C0(IR^{n}) are also proved.

1. Introduction

Some variational problems (cf. for example [3], [8], [9], [12]+[15], [12]+[14], [27]) naturally select some subsets of the Sobolev space W_{loc}^{1,p}(IR^{n}) pointwise constraints on the gradients and of which it would be interesting to look for abstract characterizations.

In a recent paper, cf. [23], some characterizations of the families of subsets of certain function spaces whose elements are subject to pointwise constraints on the gradient have been established.

For example, in the case of W^{1,p} functions, given a family {K(Ω): Ω bounded open set} of subsets of W_{loc}^{1,p}(IR^{n}), necessary and sufficient conditions on the family have been given so that

K(Ω) = {u ∈ W_{loc}^{1,p}(IR^{n}): Du(x) ∈ C for a.e. x in Ω}, for every bounded open set Ω with Lipschitz boundary.

(1.1)

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

168 A. Corbo Esposito, R. De Arcangelis / A characterization of sets of functions ...

C being a closed convex subset of IR^{n}.

On the other side, fixed an open set Ω in IR^{n} and given a single subset K(Ω) of W_{loc}^{1,p}(IR^{n}), necessary and sufficient conditions in order to characterize K(Ω) as in (1.1) generally look to be rather elaborated, unless Ω = IR^{n}.

This is just the case treated in the present paper.

In order to describe the results obtained let us first recall that for every subset E of IR^{n} the characteristic function χ_{E} of E is defined by χ_{E}(x) = 1 if x ∈ E, χ_{E}(x) = 0 if x ∈ IR^{n}\E and that a piecewise affine function u on IR^{n} is a continuous function that can be expressed as

u(x) = ∑_{j=1}^{m}(⟨z_{j}, x⟩ + s_{j})χ_{p_{j}}(x) x ∈ IR^{n}

where z_{1}, …, z_{m} ∈ IR^{n}, s_{1}, …, s_{m} ∈ IR and P_{1}, …, P_{m} are pairwise disjoint polyhedra of IR^{n} with nonempty interiors such that ⋃_{j=1}^{m} P_{j} = IR^{n}.

In the case of W^{1,p} functions with p ∈ [1, +∞[, we first observe that if C is a closed convex subset of IR^{n} and

K = {u ∈ W_{loc}^{1,p}(IR^{n}): Du(x) ∈ C for a.e. x in IR^{n}},  (1.2)

then K verifies the following conditions

u ∈ K, c ∈ IR, y ∈ IR^{n} ⟹ (i) u + c ∈ K,
(ii) u(· + y) ∈ K,
(iii) \frac{1}{t}u(t·) ∈ K;

(1.3)

K is convex;  (1.4)
K is W_{loc}^{1,p}(IR^{n}) closed;  (1.5)

u = ∑_{j=1}^{n}(⟨z_{j}, ·⟩ + s_{j})χ_{P_{j}} piecewise affine function on IR^{n}
with ⟨z_{j}, ·⟩ ∈ K for every j ∈ {1, …, m} ⇒ u ∈ K.

(1.6)

Then we prove that the above conditions are indeed sufficient in order to characterize the subsets of W_{loc}^{1,p}(IR^{n}) that can be described as in (1.2).

In fact we prove that if K is a subset of W_{loc}^{1,p}(IR^{n}), with p ∈ [1, +∞[, verifying (1.3)÷(1.6) and if C is the subset of IR^{n} defined by

C = {z ∈ IR^{n}: ⟨z, ·⟩ ∈ K},  (1.7)

then C is closed, convex and (1.2) holds.

The above result is deduced as a particular case by a more general characterization result (Theorem 6.3) holding for subsets of a Hausdorff locally convex topological vector subspace of the space D′ of the distributions on IR^{n} verifying suitable structure conditions.

As corollaries of Theorem 6.3, some characterization results for subsets of BV_{loc}(IR^{n}), of L_{loc}^{p}(IR^{n}), of the set of the Radon measures on IR^{n}, of S′(IR^{n}) and of D′(IR^{n}) are also proved.

For example we prove that if K is a subset of BV_{loc}(IR^{n}) verifying (1.3), (1.4), (1.6) and

K is w^{∗} − BV_{loc}(IR^{n}) sequentially closed,  (1.8)

then the set C in (1.7) is closed convex and

K = \{u ∈ BV_{loc}(IR^{n}):\frac{1}{meas(A)} ∫_{A} dDu ∈ C

for every nonempty bounded open subset A of IR^{n}}

(1.9)

(Theorem 7.4).

Analogously, if K is a subset of D′(IR^{n}) verifying (1.3), (1.4), (1.6) and

K is w^{∗} − D′(IR^{n}) sequentially closed,  (1.10)

then the set C in (1.7) is closed convex and

K = {u ∈ D′(IR^{n}): −⟨u, Dφ⟩ ∈ C for every φ ∈ D(IR^{n}) with φ ≥ 0 a.e., ∫_{IR_{n}} φ = 1}

(1.11)

(Theorem 7.1).

In both cases it is observed that, given a closed convex subset C of IR^{n}, the set K given by (1.9) (respectively by (1.11)) verifies conditions (1.3), (1.4), (1.6) and (1.8) (respectively (1.10)).

In conclusion we observe that our results might be applied, for example, to the study of homogenization problems in elastic-plastic torsion theory, we refer to [23] for a discussion of this topic in a context very similar to our one.

2. Notations and preliminary results

For every p ∈ [1, +∞] W_{loc}^{1,p}, respectively L_{loc}^{p}, C^{0}, denotes the space W_{loc}^{1,p}(IR^{n}), respectively L_{loc}^{p}(IR^{n}), C^{0}(IR^{n}); D denotes the space D(IR^{n}) of the C^{∞} functions with compact support in IR^{n} and D′ the space D′(IR^{n}) of the distributions on IR^{n}.

w^{∗} − W_{loc}^{1,∞}(IR^{n}) (weak^{∗} − W_{loc}^{1,∞}(IR^{n})) denotes the projective limit topology on W_{loc}^{1,∞} of the spaces W^{1,∞}(Ω), Ω bounded open set, endowed with their w^{∗} − W^{1,∞}(Ω) topologies with respect to the embedding mappings of W_{loc}^{1,∞} in W^{1,∞}(Ω).

In a similar manner the topologies W_{loc}^{1,p}(IR^{n}), w −W_{loc}^{1,p}(IR^{n}) (weak-W_{loc}^{1,p}(IR^{n})), L_{loc}^{p}(IR^{n}), w − L_{loc}^{p}(IR^{n}) (weak-L_{loc}^{p}(IR^{n})) and w^{∗} − L_{loc}^{∞}(IR^{n}) (weak^{∗} − L_{loc}^{∞}(IR^{n})) are defined.

Moreover C_{loc}^{0}(IR^{n}) denotes the projective limit topology on C^{0} of the spaces C^{0}(Ω̄), Ω bounded open set, endowed with their uniform convergence topologies with respect to the embedding mappings of C^{0} in C^{0}(Ω̄).

As usual S denotes the set of the rapidly decreasing functions, C_{0}^{0} the set of the continuous functions on IR^{n} with compact support, C_{0}^{1} the set of the continuously differentiable functions on IR^{n} with compact support, S′ the set of the tempered distributions and M_{loc} the set of the Radon measures on IR^{n}, i.e. the set of the regular measures defined at least for every Borel subset of IR^{n} and with locally finite variations.

D(IR^{n}), respectively S(IR^{n}), denotes the usual topology of D, respectively of S, and w^{∗} − D′(IR^{n}), respectively w^{∗} − S′(IR^{n}), the weak^{∗} one of D′, respectively of S′. Moreover C_{0}^{0}(IR^{n}) denotes the usual strict inductive limit topology on C_{0}^{0} that makes it a LF-space; we recall that, by using Riesz representation theorem (see e.g. [42] 7.18), the dual space of C_{0}^{0}, endowed with the C_{0}^{0}(IR^{n}) topology, is isomorphic to M_{loc}. The weak^{∗} topology of M_{loc} is denoted by w^{∗} − M_{loc}(IR^{n}).

Let Ω be an open set, by BV (Ω) we denote the set of the functions in L^{1}(Ω) having distributional partial derivatives that are Radon measures with finite total variations on Ω.

w^{∗} − BV (Ω) denotes the weak^{∗} − BV (Ω) topology on BV (Ω) (cf. e.g. [23] for a precise description of this topology).

BV_{loc} denotes the set of functions in L_{loc}^{1} that are in BV (A) for every bounded open set A and w^{∗} − BV_{loc}(IR^{n}) the projective limit topology on BV_{loc} of the spaces BV (Ω), Ω bounded open set, endowed with their w^{∗}−BV (Ω) topology with respect to the embedding mappings of BV_{loc} in BV (Ω).

For a wide exposition about BV functions we refer to [36], here we only recall that BV_{loc}, endowed with its w^{∗} − BV_{loc}(IR^{n}) topology, is a sequentially complete Hausdorff locally convex topological vector space.

For every measurable subset E of IR^{n} |E| denotes the Lebesgue measure of E and, for every z in IR^{n}, u_{z} the linear function u_{z}(x) = ⟨z, x⟩, x ∈ IR^{n}.

Furthermore we recall that a subset of IR^{n} is said to be a polyhedron if it is the intersection of a finite number of half-spaces.

Let (U, τ) be a topological space and let X be a subset of U; (U, τ) − cl(X) denotes the closure of X in U, i.e. the set of the points in U that can be approximated by generalized sequences in X, and by (U, τ)_{seq} − cl(X) the sequential closure of X in U, i.e. the set of the points in U that can be approximated by sequences in X. Obviously (U, τ)_{seq} − cl(X) ⊆ (U, τ) − cl(X) for every subset X of U. Let V be a vector space and let S be a subset of V, conv(S) denotes the convex hull of S, i.e. the set of the finite convex combinations of points of S. Given a subset C of IR^{n}, Σ(C) denotes the affine hull of C, that is the smallest affine subset of IR^{n} containing C.

We now recall the notion of integral of a function taking values in a topological vector space (cf. [41]). Let (U, τ) be a Hausdorff locally convex topological vector space and let {p_{a}}_{a∈A} be a family of seminorms defining τ. Let E be a Lebesgue measurable subset of IR^{n} and let f be a function from E to U.

Definition 2.1. The function f is said to be τ-integrable on E if there exists u ∈ U such that for every η > 0 and a ∈ A there exists a partition Δ_{η,a} = {B_{η,a,j}}_{j=1,...,m} of E into measurable sets such that

sup {p_{a} (∑_{j=1}^{m} f(y_{j})|B_{η,a,j}| − u): y_{j} ∈ B_{η,a,j}, j ∈ {1, …, m}} < η.  (2.1)

The vector u is the value of the integral of f on E and is denoted by ∫_{E} f(y)dy.

We recall a property about the behaviour of the integral of a vector valued function with respect to duality (cf. [41] Corollary 5.2).

Theorem 2.2. Let f: E → U be τ-integrable on E and let L ∈ U^{∗}. Then ⟨L, f(·)⟩ is Lebesgue integrable on E and ∫_{E}⟨L, f(y)⟩dy = ⟨L, ∫_{E} f(y)dy⟩.

By Theorem 2.2 and well known subdifferentiability properties of convex functions (cf. for example [34]), the following Jensen type inequality is soon deduced.

Proposition 2.3. Let E be a Lebesgue measurable subset of IR^{n} with nonzero finite measure; let f: E → U be τ-integrable and such that for every open subset A of U f^{−1}(A) is Lebesgue measurable. Let Φ: U → [0, +∞[ be convex, then Φ ∘ f is Lebesgue measurable on E and

Φ (\frac{1}{|E|} ∫_{E} f(y)dy) ≤ \frac{1}{|E|} ∫_{E} Φ(f(y))dy.  (2.2)

The following result yields an integrability condition (cf. [23] Proposition 2.4).

Proposition 2.4. Let E be a Lebesgue measurable subset of IR^{n}, let (U, τ) be a sequentially complete Hausdorff locally convex topological vector space and let f: E → U be continuous with compact support. Then f is τ-integrable on E and ∫_{E} f(y)dy is the limit of the sequences of the Cauchy sums of f, i.e. ∫_{E} f(y)dy = lim_{h} ∑_{j=1}^{m_{h}} |Q_{j}^{h} ∩ E|f(y_{j}^{h}) whenever, for every h ∈ IN, {Q_{j}^{h}}_{j}

is a partition of IR^{n} made up by half open cubes with sidelength \frac{1}{h}, spt(f) ∩ Q_{j}^{h} ≠ ∅ if and

only if j ∈ {1, …, m_{h}} and y_{1}^{h} ∈ Q_{1}^{h} ∩ E, …, y_{m_{h}} ∈ Q_{m_{h}}^{h} ∩ E.

For every distribution u in D′, y ∈ IR^{n}, t > 0 the translation of u by the vector y T [y]u and the homothety of u by coefficient t O′_{t}u are the distributions defined by (cf. [39], Chapter 1, §2.6).

⟨T [y]u, φ⟩ = ⟨u, φ(· − y)⟩ φ ∈ D,  (2.3)

⟨O′_{t}u, φ⟩ = \frac{1}{t^{n}} ⟨u, φ (\frac{1}{t}·)⟩ φ ∈ D.  (2.4)

In the following it will be useful to introduce the “rescaled” homothety O_{t}u of u by coefficient t defined by

O_{t}u = \frac{1}{t}O′_{t}u.  (2.5)

Obviously, in the case of real functions on IR^{n}, T [y] and O_{t} turn out to be the operators defined on the function u by T [y]u = u(· + y), O_{t}u = \frac{1}{t}u(t·). For every r > 0 B_{r} denotes the open ball of IR^{n} centered at the origin and of radius r. Let α be a symmetric mollifier, i.e. a nonnegative function in C^{∞}(IR^{n}) such that spt(α) ⊆ B_{1}, ∫_{IR}^{n} α = 1 and α(−x) = α(x) for every x in IR^{n}. For every ϵ > 0 and x in IR^{n} set α^{(ϵ)}(x) = ϵ^{−n}α(x/ϵ) and define, for every distribution u in D′, the regularization u_{ϵ} of u by

u_{ϵ}(x) = (u ∗ α^{(ϵ)})(x) = ⟨u, α^{(ϵ)}(x − ·)⟩ x ∈ IR^{n}.  (2.6)

Obviously if u ∈ L_{loc}^{1}, the regularization u_{ϵ} of u turns out to be given by the classical formula

u_{ϵ}(x) = ∫_{IR_{n}} α^{(ϵ)}(x − y)u(y)dy x ∈ IR^{n}.  (2.7)

It is well known, see for example [47] VI.3, that for every u ∈ D′

u_{ϵ} ∈ C^{∞}(IR^{n}), D^{θ}u_{ϵ} = u ∗ (D^{θ}α^{(ϵ)}) = (D^{θ}u) ∗ α^{(ϵ)} for every ϵ > 0 and every multiindex θ ∈ (IN ∪ {0})^{n};

(2.8)

u_{ϵ} → u in w^{∗} − D′(IR^{n}) as ϵ → 0^{+}  (2.9)

and

⟨u_{ϵ}, φ⟩ = ⟨u, φ_{c}⟩ for every φ ∈ D, ϵ > 0.  (2.10)

We also recall that the regularization of a distribution can be described by means of the integral of its translations.

Proposition 2.5. Let α be the mollifier appearing in (2.6). Then for every u ∈ D′, ϵ > 0 the function y ∈ IR^{n} ↦ α(y)T [ϵy]u ∈ D′ is w^{∗} − D′(IR^{n}) integrable on IR^{n}, the integral ∫_{IR}^{n} α(y)T [ϵy]u dy is indeed a function and

(∫_{IR_{n}} α(y)T [ϵy]u dy) (x) = u_{ϵ}(x) for a.e. x in IR^{n}.  (2.11)

In order to get informations about the left hand side of (2.11) let us consider a Hausdorff locally convex topological vector subspace (U, τ) of D′ such that

u ∈ U, y ∈ IR^{n} ⇒ T [y]u ∈ U.  (2.12)

Proposition 2.6. Let (U, τ) be a sequentially complete Hausdorff locally convex topological vector subspace of D′ verifying (2.12). Let α be the mollifier appearing in (2.6), u ∈ U and ϵ > 0; assume that spt(α) is convex and that the function y ∈ IR^{n} ↦ T [y]u ∈ U is τ-continuous on IR^{n}. Then it results

∫_{IR_{n}} α(y)T [ϵy]u dy ∈ (U, τ)_{seq} − cl(conv({T [y]u, y ∈ B_{ϵ}})).  (2.13)

Proof. For every h ∈ IN let {Q_{j}^{h}}_{j∈IN} be a sequence of pairwise disjoint half open cubes

of IR^{n} with sidelength \frac{1}{h} and ⋃_{j=1}^{h} Q_{j}^{h} = IR^{n} such that for some m_{h} ∈ IN Q_{j}^{h} ∩ spt(α) ≠ ∅ if and only if j ∈ {1, …, m_{h}}. Since for every j ∈ {1, …, m_{h}} the set Q_{j}^{h} ∩ spt(α) is connected, let y_{j}^{h} ∈ Q_{j}^{h} ∩ spt(α) be such that

∫_{Q_{j}^{h}∩spt(α)} α(y)dy = α(y_{j}^{h})|Q_{j}^{h} ∩ spt(α)|,  (2.14)

then by Proposition 2.4 we get that the integral ∫_{IR}^{n} α(y)T [ϵy]u dy exists and that

∑_{j=1}^{m_{h}} |Q_{j}^{h} ∩ spt(α)|α(y_{j}^{h})T [ϵy_{j}^{h}]u → ∫_{IR_{n}} α(y)T [ϵy]u dy.  (2.15)

By (2.14), once observed that ∑_{j=1}^{m_{h}} ∫_{Q_{j}^{h}∩spt(α)} α(y)dy = 1, we soon deduce that

∑_{j=1}^{m_{h}} |Q_{j}^{h} ∩ spt(α)|α(y_{j}^{h})T [ϵy_{j}^{h}]u =

= ∑_{j=1}^{m_{h}} (∫_{Q_{j}^{h}∩spt(α)} α(y)dy) T [ϵy_{j}^{h}]u ∈ conv{T [y]u, y ∈ B_{ϵ}},

(2.16)

hence by (2.16) and (2.15) condition (2.13) follows.

In conclusion we recall the following results, see [23] Proposition 4.3 and Proposition 4.4.

Proposition 2.7. Let C be a convex subset of IR^{n} and let ψ be a function in (L_{loc}^{1})^{n} such that

ψ(x) ∈ C for a.e. x in IR^{n}.  (2.17)

Then for every ϵ > 0 it results

ψ_{ϵ}(x) ∈ C̄ for every x in IR^{n}.  (2.18)

Proposition 2.8. Let G be a convex subset of IR^{m} such that ∂G ≠ ∅ and B_{r} ⊆ G^{∘} for some r > 0. Let t ∈]0, 1[, then

dist(tG, ∂G) ≥ r(1 − t).  (2.19)

3. Distributions with constraints on the gradient

Let C be a subset of IR^{n}. In this section we study some properties of the sets of the distributions on IR^{n} verifying a constraint on the gradient determined by C.

Such sets are defined by

K_{C} = {u ∈ D′: −⟨u, Dφ⟩ ∈ C for every φ ∈ D with φ ≥ 0, ∫_{IR_{n}} φ = 1}.  (3.1)

In order to describe in some significant cases the sets in (3.1) let us introduce, for every p ∈ [1, +∞], the following sets

K_{C}^{1,p} = {u ∈ W_{loc}^{1,p}: Du(x) ∈ C for a.e. x in IR^{n}},  (3.2)

K_{C}^{BV} =\{u ∈ BV_{loc}: \frac{1}{|A|} ∫_{A} dDu ∈ C

for every nonempty bounded open subset A of IR^{n}},

(3.3)

K_{C}^{0} ={u ∈ C^{0}: − ∫_{IR_{n}} u dDφ ∈ C
for every φ ∈ BV_{loc} with compact support, φ ≥ 0 a.e., ∫_{IR_{n}} φ = 1}

(3.4)

and, being p′ the conjugate exponent of p,

K_{C}^{p} ={u ∈ L_{loc}^{p}: − ∫_{IR_{n}} uDφ ∈ C
for every φ ∈ W_{loc}^{1,p′} with compact support, φ ≥ 0 a.e., ∫_{IR_{n}} φ = 1}.

(3.5)

Moreover we set

K_{C}^{M} ={u ∈ M_{loc}: − ∫_{IR_{n}} Dφdu ∈ C
for every φ ∈ C_{0}^{1} with compact support, φ ≥ 0, ∫_{IR_{n}} φ = 1}

(3.6)

and

K_{C}^{S} ={u ∈ S′: ⟨Du, φ⟩ ∈ C
for every φ ∈ S with φ ≥ 0, ∫_{IR_{n}} φ = 1}.

(3.7)

We have Proposition 3.1. Let C be a closed convex subset of IR^{n}, p ∈ [1, +∞] and let K_{C} and K_{C}^{1,p} be defined respectively by (3.1) and (3.2). Then

K_{C}^{1,p} = K_{C} ∩ W_{loc}^{1,p}.  (3.8)

Proof. Let u be in K_{C} ∩ W_{loc}^{1,p}, then by the divergence theorem we have

∫_{IR_{n}} φDu ∈ C for every φ ∈ D with φ ≥ 0, ∫_{IR_{n}} φ = 1,  (3.9)

whence we soon deduce that

K_{C} ∩ W_{loc}^{1,p} ⊆ K_{C}^{1,p}.  (3.10)

On the other side, being C closed and convex, there exists a family F = {(a_{σ}, b_{σ})}_{σ∈S} ⊆ IR^{n} × IR such that

z ∈ C ⇔ ⟨a_{σ}, z⟩ + b_{σ} ≥ 0 for every σ ∈ S,  (3.11)

therefore by (3.11) we get for every u in K_{C}^{1,p}

⟨a_{σ}, Du(x)⟩ + b_{σ} ≥ 0 for every σ ∈ S, and a.e. x ∈ IR^{n}.  (3.12)

Let now φ be in D with φ ≥ 0, ∫_{IR}^{n} φ = 1, then by multiplying both sides of (3.12) by φ(x), integrating over IR^{n} and the divergence theorem we get

− ⟨a_{σ}, ∫_{IR_{n}} uDφ⟩ + b_{σ} ≥ 0 for every σ ∈ S.  (3.13)

By (3.13) and (3.11) we deduce

K_{C}^{1,p} ⊆ K_{C} ∩ W_{loc}^{1,p},  (3.14)

hence by (3.10) and (3.14) equality (3.8) follows.

Proposition 3.2. Let C be a closed and convex subset of IR^{n} and let K_{C} and K_{C}^{BV} be defined respectively by (3.1) and (3.3). Then

K_{C}^{BV} = K_{C} ∩ BV_{loc}.  (3.15)

Proof. Let u be in K_{C} ∩ BV_{loc}. Let A be a nonempty bounded open set and let {φ_{h}}_{h} ⊆ C_{0}^{∞}(A) be such that 0 ≤ φ_{h} ≤ φ_{h+1} and φ_{h}(x) → χ_{A}(x) for every x ∈ IR^{n}. Being u in BV_{loc}, by the closedness of C we get

\frac{1}{|A|} ∫_{A} dDu = lim_{h} \frac{1}{|A|} ∫_{IR_{n}} φ_{h}dDu = − lim_{h} ∫_{IR_{n}} udD (\frac{1}{∫_{A} φ_{h}} φ_{h}) ∈ C  (3.16)

hence by (3.16) we deduce

K_{C} ∩ BV_{loc} ⊆ K_{C}^{BV} .  (3.17)

In order to prove the reverse inclusion in (3.17) we first observe that if u ∈ K_{C}^{BV} then by approximation it follows that

\frac{1}{|E|} ∫_{E} dDu ∈ C for every bounded Borel set E such that |E| > 0.  (3.18)

Let u ∈ K_{C}^{BV}, ψ ∈ D be such that ψ ≥ 0, ν ∈ IN; let us consider a partition of IR^{n} made up by half open cubes Q_{j}^{ν}, j ∈ IN, of sidelength \frac{1}{ν} and let S_{ν} = \{j ∈ IN: Q_{j}^{ν} ∩ spt(ψ) ≠ ∅\},

then by (3.18) we have that \frac{1}{|Q_{j}^{ν}|} ∫_{Q_{j}^{ν}} dDu ∈ C for every j ∈ S_{ν}, hence, being C convex,

we have

\frac{1}{∫_{IR}_{n} ψ} ∫_{IR_{n}} ∑_{j∈S_{ν}}(\frac{1}{|Q_{j}^{ν}|} ∫_{Q_{j}^{ν}} ψ)χ_{Q_{j}^{ν}}dDu =

= ∑_{j∈S_{ν}} \frac{∫_{Q_{j}^{ν}} ψ}{∫_{IR}_{n} ψ} \frac{1}{|Q_{j}^{ν}|} ∫_{Q_{j}^{ν}} dDu ∈ C for every ν ∈ IN, j ∈ S_{ν}.

(3.19)

Let us now observe that the sequence of functions ∑_{j∈S_{ν}} (\frac{1}{|Q_{j}^{ν}|} ∫_{Q_{j}^{ν}} ψ) χ_{Q_{j}^{ν}} converges

uniformly on IR^{n} to ψ as ν → ∞, hence, being Du a measure with finite total variation on spt(ψ), by taking the limit as ν → ∞ in (3.19) we obtain

−⟨u, Dφ⟩ = ∫_{IR_{n}} φdDu ∈ C for every φ ∈ D with φ ≥ 0, ∫_{IR_{n}} φ = 1,  (3.20)

that is

K_{C}^{BV} ⊆ K_{C} ∩ BV_{loc}.  (3.21)

By (3.17) and (3.21) equality (3.15) follows.

Proposition 3.3. Let C be a closed and convex subset of IR^{n} and let K_{C}, K_{C}^{0}, K_{C}^{p}, p ∈ [1, +∞], and K_{C}^{M} be defined respectively by (3.1), (3.4), (3.5) and (3.6). Then

K_{C}^{0} = K_{C} ∩ C^{0};  (3.22)

K_{C}^{p} = K_{C} ∩ L_{loc}^{p} for every p ∈ [1, +∞];  (3.23)

K_{C}^{M} = K_{C} ∩ M_{loc}.  (3.24)

Proof. Let us prove (3.22), the proof of (3.23) and (3.24) being similar with obvious changes.

It is clear that

K_{C}^{0} ⊆ K_{C} ∩ C^{0}.  (3.25)

In order to prove the reverse inclusion let u be in K_{C} ∩ C^{0} and let φ ∈ BV (IR^{n}) with compact support, φ ≥ 0 a.e. and ∫_{IR}^{n} φ = 1. For every ϵ > 0 let φ_{ϵ} be the regularization of φ given by (2.7), then φ_{ϵ} ∈ D, φ_{ϵ} ≥ 0 in IR^{n}, ∫_{IR}^{n} φ_{ϵ} = 1 and ∫_{IR}^{n} ψDφ_{ϵ} → ∫_{IR}^{n} ψdDφ for every ψ in C^{0}. Being u in K_{C} ∩ C^{0} we have

− ∫_{IR_{n}} uDφ_{ϵ} ∈ C for every ϵ > 0,  (3.26)

hence by the closedness of C, as ϵ → 0^{+}, we deduce by (3.26) that

K_{C} ∩ C^{0} ⊆ K_{C}^{0}.  (3.27)

By (3.25) and (3.27) equality (3.22) follows.

Proposition 3.4. Let C be a closed and convex subset of IR^{n} and let K_{C} and K_{C}^{S} be defined respectively by (3.1) and (3.7). Then

K_{C}^{S} = K_{C} ∩ S′.  (3.28)

Proof. Let us recall that (see e.g. [47] VI.1)

\{ \begin{matrix}u ∈ S′ ⇒ Du ∈ (S′)^{n} \\ ⟨u, Dφ⟩ = −⟨Du, φ⟩ & for every u ∈ S′, φ ∈ S, \end{matrix}  (3.29)

whence we soon have

K_{C}^{S} ⊆ K_{C} ∩ S′.  (3.30)

In order to prove the reverse inclusion in (3.30) let {ψ_{h}}_{h} ⊆ D be such that 0 ≤ ψ_{h} ≤ 1, ψ_{h}(x) = 1, if |x| ≤ h and sup_{x∈IR^{n}} |D^{θ}ψ_{h}(x)| ≤ 1 for every h ∈ IN and every multindex θ ∈ IN^{n}, then it is easy to see that

φψ_{h} → φ in S(IR^{n}) for every φ ∈ S.  (3.31)

Let u ∈ K_{C} ∩ S′, φ ∈ S with φ ≥ 0 and ∫_{IR}^{n} φ = 1, then by (3.29) and the closedness of C we deduce that

⟨Du, φ⟩ = −⟨u, Dφ⟩ = − lim_{h} ⟨u, D (\frac{1}{∫_{IR}^{n} ψ_{h}φ}ψ_{h}φ)⟩ ∈ C  (3.32)

and

K_{C} ∩ S′ ⊆ K_{C}^{S}.  (3.33)

By (3.30) and (3.33) equality (3.28) follows. We now go back to the general case. Proposition 3.5. Let C be a subset of IR^{n} and let K_{C} be given by (3.1). Then

u ∈ K_{C}, c ∈ IR, y ∈ IR^{n}, t > 0 ⟹ (i) u + c ∈ K_{C},
(ii) T [y]u ∈ K_{C},
(iii) O_{t}u ∈ K_{C};

(3.34)

C convex ⇒ K_{C} convex;  (3.35)
C closed ⇒ K_{C} w^{∗} − D′(IR^{n}) closed;  (3.36)

u = ∑_{j=1}^{m}(u_{z_{j}} + s_{j})χ_{P_{j}} piecewise affine function on IR^{n}

such that u_{z_{j}} ∈ K_{C} for every j ∈ {1, …, m} ⇒ u ∈ K_{C}.

(3.37)

Proof. Conditions (i) and (ii) in (3.34) can be easily obtained. Let us prove (iii) of (3.34).

Let u ∈ K_{C}, t > 0 and let φ ∈ D with φ ≥ 0 and ∫_{IR}^{n} φ = 1. Let us observe that

O_{1/t}(Dφ) = tD(O_{1/t}φ),  (3.38)

hence by (3.38) we have

\begin{aligned}−⟨O_{t}u, Dφ⟩ & = −\frac{1}{t^{n+2}} ⟨u, O_{1/t}(Dφ)⟩ = \\ & = −\frac{1}{t^{n+1}} ⟨u, D (O_{1/t}φ)⟩ = \\ & = −⟨u, D (\frac{1}{t^{n+1}} O_{1/t}φ)⟩. \end{aligned}  (3.39)

A. Corbo Esposito, R. De Arcangelis / A characterization of sets of functions ... 179

By (3.39), once observed that the function \frac{1}{t^{n+1}} O_{1/t}φ is in D with \frac{1}{t^{n+1}} O_{1/t}φ ≥ 0, ∫_{IR}^{n} \frac{1}{t^{n+1}} O_{1/t}φ = 1 and once recalled that u is in K_{C}, we get that

−⟨O_{t}u, Dφ⟩ ∈ C for every φ ∈ D with φ ≥ 0 and ∫_{IR_{n}} φ = 1.  (3.40)

By (3.40) condition (iii) in (3.34) follows. Condition (3.35) is trivially verified. Let us prove (3.36). To this aim let u ∈ D′ and let {u_{λ}}_{λ∈Λ} be a generalized sequence in D′ that converges to u in w^{∗} − D′(IR^{n}). For every φ ∈ D with φ ≥ 0 and ∫_{IR}^{n} φ = 1, by using the closedness of C, it follows that

−⟨u, Dφ⟩ = lim_{λ} −⟨u_{λ}, Dφ⟩ ∈ C  (3.41)

and (3.36) comes. In conclusion let us prove (3.37). Let u = ∑_{j=1}^{m}(u_{z_{j}} + s_{j})χ_{p_{j}} be as in (3.37), then by Proposition 3.1 applied with p = +∞ it follows that

z_{j} ∈ C for every j ∈ {1, …, m},  (3.42)

therefore by (3.42) we deduce

Du(x) ∈ C for a.e. x in IR^{n}.  (3.43)

By (3.43) and again Proposition 3.1 applied with p = +∞, (3.37) follows.

4. A representation result Let C be a closed convex set in IR^{n} and (U, τ) a topological vector subspace of D′, in the present section we want to prove a representation result for the set (U, τ)_{seq} − cl(K_{C}^{1,∞}), K_{C}^{1,∞} being defined by (3.2) with p = +∞. We assume that

τ is finer than w^{∗} − D′(IR^{n})  (4.1)

and that, being for u ∈ D′ and ϵ > 0, u_{ϵ} the regularization of u given by (2.7),

\{ \begin{matrix}(i) & u_{ϵ} ∈ U for every u ∈ U, ϵ > 0 small enough, \\ (ii) & u_{ϵ} → u in τ as ϵ → 0^{+} for every u ∈ U. \end{matrix}  (4.2)

Proposition 4.1. Let (U, τ) be a topological vector subspace of D′ verifying (4.1) and (4.2).

Let C be a closed convex subset of IR^{n}, K_{C}^{1,∞} be defined by (3.2) with p = +∞ and K_{C} by (3.1). Then

(U, τ) − cl(K_{C}^{1,∞} ∩ U) = (U, τ)_{seq} − cl(K_{C}^{1,∞} ∩ U) = K_{C} ∩ U.  (4.3)

Proof. Let us preliminarily observe that by Proposition 3.1 with p = +∞ we have

K_{C}^{1,∞} = K_{C} ∩ W_{loc}^{1,∞} ⊆ K_{C}.  (4.4)

By Proposition 3.5 the set K_{C} is w^{∗}−D′(IR^{n}) closed, therefore by (4.1) K_{C} ∩U is τ-closed. From this and (4.4) we infer

(U, τ) − cl(K_{C}^{1,∞} ∩ U) ⊆ K_{C} ∩ U.  (4.5)

In order to prove the reverse inclusion in the sequential case, let u be in K_{C} ∩ U and let, for ϵ > 0, u_{ϵ} be the regularization of u given by (2.7).

For fixed ϵ > 0 and x in IR^{n} the function α^{(ϵ)}(x − ·) is in D with α^{(ϵ)}(x − ·) ≥ 0, ∫_{IR}^{n} α^{(ϵ)}(x − y)dy = 1 and

(Dα^{(ϵ)})(x − y) = −D(α^{(ϵ)}(x − ·))(y) for every y ∈ IR^{n},  (4.6)

the gradient in the left hand side of (4.6) being taken with respect to the set of the variables of α^{(ϵ)}; therefore by (2.8) and (4.6) it results

Du_{ϵ}(x) = D(u ∗ α^{(ϵ)})(x) = ⟨u, (Dα^{(ϵ)})(x − ·)⟩ = −⟨u, D(α^{(ϵ)}(x − ·))⟩ ∈ C  (4.7)

that is, by (4.2) (i),

u_{ϵ} ∈ K_{C}^{1,∞} ∩ U for every ϵ > 0 small enough.  (4.8)

As ϵ → 0^{+} by (4.2) we deduce that

K_{C} ∩ U ⊆ (U, τ)_{seq} − cl(K_{C}^{1,∞} ∩ U).  (4.9)

By (4.5) and (4.9) the thesis follows.

By applying Proposition 4.1 to some particular cases we obtain the following results.

Proposition 4.2. Let C be a closed convex subset of IR^{n}. For every p ∈ [1, +∞] let K_{C}^{p} be defined by (3.5), K_{C}^{0} by (3.4) and K_{C}^{1,∞} by (3.2). Then

(L_{loc}^{p}, L_{loc}^{p}(IR^{n}))−cl(K_{C}^{1,∞}) = (L_{loc}^{p}, L_{loc}^{p}(IR^{n}))_{seq} −cl(K_{C}^{1,∞}) = K_{C}^{p} if p ∈ [1, +∞[; (4.10)

(L_{loc}^{∞}, w^{∗} − L_{loc}^{∞}(IR^{n})) − cl(K_{C}^{1,∞}) = (L_{loc}^{∞}, w^{∗} − L_{loc}^{∞}(IR^{n}))_{seq} − cl(K_{C}^{1,∞}) = K_{C}^{∞}; (4.11)

(C^{0}, C_{loc}^{0}(IR^{n})) − cl(K_{C}^{1,∞}) = (C^{0}, C_{loc}^{0}(IR^{n}))_{seq} − cl(K_{C}^{1,∞}) = K_{C}^{0}.  (4.12)

Proof. Let us prove (4.12), the proof of (4.10) and (4.11) being similar with the obvious changes of spaces. We observe that the space (C^{0}, C_{loc}^{0}(IR^{n})) satisfies conditions (4.1) and (4.2), hence by Proposition 4.1, once observed that K_{C}^{1,∞} ∩ C^{0} = K_{C}^{1,∞}, we get that

(C^{0}, C_{loc}^{0}(IR^{n})) − cl(K_{C}^{1,∞}) = (C^{0}, C_{loc}^{0}(IR^{n}))_{seq} − cl(K_{C}^{1,∞}) =
= {u ∈ C^{0}: − ∫_{IR_{n}} uDφ ∈ C for every φ ∈ D with φ ≥ 0, ∫_{IR_{n}} φ = 1} .

(4.13)

Therefore the thesis follows by (4.13) and Proposition 3.3, although the first equality in (4.12) can be deduced by standard general topology arguments.

Proposition 4.3. Let C be a closed convex subset of IR^{n}. For every p ∈ [1, +∞] let K_{C}^{1,p} be defined by (3.2) and K_{C}^{BV} by (3.3). Then

(BV_{loc}, w^{∗} − BV_{loc}(IR^{n})) − cl(K_{C}^{1,∞}) = (BV_{loc}, w^{∗} − BV_{loc}(IR^{n}))_{seq} − cl(K_{C}^{1,∞}) = K_{C}^{BV}; (4.14)

(W_{loc}^{1,p}, W_{loc}^{1,p}(IR^{n}))−cl(K_{C}^{1,∞}) = (W_{loc}^{1,p}, W_{loc}^{1,p}(IR^{n}))_{seq} −cl(K_{C}^{1,∞}) = K_{C}^{1,p} if p ∈ [1, +∞[; (4.15)

(W_{loc}^{1,∞}, w^{∗} − W_{loc}^{1,∞}(IR^{n})) − cl(K_{C}^{1,∞}) = (W_{loc}^{1,∞}, w^{∗} − W_{loc}^{1,∞}(IR^{n}))_{seq} − cl(K_{C}^{1,∞}) = K_{C}^{1,∞}. (4.16)

Proof. Let us prove (4.14). We observe that (BV_{loc}, w^{∗} − BV_{loc}(IR^{n})) satisfies conditions (4.1) and (4.2) and that K_{C}^{1,∞} ∩ BV_{loc} = K_{C}^{1,∞}, hence (4.14) follows by Proposition 4.1 and Proposition 3.2. The proof of (4.15) and (4.16) are similar to the one of (4.14) with the obvious changes and by using Proposition 3.1 in place of Proposition 3.2.

5. The characterization result in a particular case

Let K be a subset of D′, in the present section and in the next one we give necessary and sufficient conditions on K for the existence of a closed convex subset C of IR^{n} such that K = K_{C}, K_{C} being defined by (3.1). Having in mind Proposition 3.5 we assume that K satisfies the following assumptions

u ∈ K, c ∈ IR, y ∈ IR^{n}, t > 0 ⟹ (i) u + c ∈ K,
(ii) T [y]u ∈ K,
(iii) O_{t}u ∈ K;

(5.1)

K convex;  (5.2)

u = ∑_{j=1}^{m}(u_{z_{j}} + s_{j})χ_{P_{j}} piecewise affine function on IR^{n} such that u_{z_{j}} ∈ K
for every j ∈ {1, …, m} ⇒ u ∈ K.

(5.3)

Let us now observe that, given a closed convex set C in IR^{n}, by Proposition 3.1 with p = +∞ it soon follows that

z ∈ C ⇔ u_{z} ∈ K_{C},  (5.4)

hence it is natural to deduce by the set K the constraint C in the following way

C = {z ∈ IR^{n}: u_{z} ∈ K}.  (5.5)

Proposition 5.1. Let K be a subset of D′ and let C be defined by (5.5). Then

a) if

K ∩ W_{loc}^{1,∞} is W_{loc}^{1,∞}(IR^{n}) closed,  (5.6)

C is closed;

b) if (5.2) holds, C is convex.

Proof. Let z ∈ IR^{n} and let {z_{h}}_{h} ⊆ C with z_{h} → z, then u_{z_{h}} ∈ K ∩ W_{loc}^{1,∞} for every h ∈ IN and

u_{z_{h}} → u_{z} in W_{loc}^{1,∞}(IR^{n}).  (5.7)

By (5.6) and (5.7) we soon deduce that u_{z} ∈ K, that is z ∈ C; by virtue of this the closedness of C follows. Finally the convexity of C trivially follows by (5.2).

In order to verify that K is indeed a set of distributions defined by the constraint on the gradient described by C in (5.5), we assume that

K ∩ W_{loc}^{1,∞} is w^{∗} − W_{loc}^{1,∞}(IR^{n}) sequentially closed.  (5.8)

Proposition 5.2. Let K be a subset of D′ verifying (5.1), (5.8); let C be defined by (5.5) and K_{C}^{1,∞} by (3.2) with p = +∞. Then

K ∩ W_{loc}^{1,∞} ⊆ K_{C}^{1,∞}.  (5.9)

Proof. Let u ∈ K ∩ W_{loc}^{1,∞}, let us prove that

Du(x_{0}) ∈ C for a.e. x_{0} ∈ IR^{n}.  (5.10)

For every x_{0} ∈ IR^{n}, t > 0 by (5.1) we have

T [−x_{0}]O_{t}T [x_{0}]u − \frac{u(x_{0})}{t} = \frac{u(x_{0} + t(· − x_{0})) − u(x_{0})}{t} ∈ K ∩ W_{loc}^{1,∞}.  (5.11)

Being u in W_{loc}^{1,∞}, by Rademacher theorem, u is differentiable a.e. in IR^{n}, i.e.

\frac{u(x_{0} + t(· − x_{0})) − u(x_{0})}{t} → ⟨Du(x_{0}), · − x_{0}⟩ in C_{loc}^{0}(IR^{n})

as t → 0^{+} for a.e. x_{0} ∈ IR^{n};

(5.12)

moreover, if x_{0} is a Lebesgue point for Du, we have

D (T [−x_{0}]O_{t}T [x_{0}]u − \frac{u(x_{0})}{t}) → Du(x_{0}) in w^{∗} − L_{loc}^{∞}(IR^{n}) as t → 0^{+}.  (5.13)

By (5.12) and (5.13) we get

T [−x_{0}]O_{t}T [x_{0}]u − \frac{u(x_{0})}{t} → ⟨Du(x_{0}), · − x_{0}⟩ in w^{∗} − W_{loc}^{1,∞}(IR^{n})

as t → 0^{+} for a.e. x_{0} ∈ IR^{n},

(5.14)

hence by (5.14), (5.11) and (5.8) we deduce that

⟨Du(x_{0}), · − x_{0}⟩ ∈ K ∩ W_{loc}^{1,∞} for a.e. x_{0} ∈ IR^{n}.  (5.15)

By (5.15) and (i) of (5.1) condition (5.10) soon follows. Finally by (5.10) we deduce the thesis.

The remaining part of this section is devoted to the proof of the reverse inclusion in (5.9). A first step in this direction is made by using condition (5.3), in fact if K is a subset of D′ verifying (5.3), C is defined by (5.5) and K_{C}^{1,∞} by (3.2) with p = +∞, then

u piecewise affine function on IR^{n} u ∈ K_{C}^{1,∞} ⇒ u ∈ K ∩ W_{loc}^{1,∞}.  (5.16)

In order to extend (5.16) to wider classes of functions let us assume that K verifies (5.1)÷(5.3) and (5.6), then, by virtue of Proposition 5.1, the set C defined in (5.5) is closed and convex. Let Σ(C) be the affine hull of C and let ν (≤ n) be its dimension, then C possesses interior points in the topology of Σ(C) and, if C ≠ ∅, it is not restrictive to assume that 0 ∈ C (in fact if this is not the case it is sufficient to consider, for z_{0} ∈ C, the sets K − u_{z_{0}} and C − z_{0}). Moreover, by using the same argument as before, it is not restrictive to assume that

0 ∈ C^{∘} the interior being taken in the topology of Σ(C).  (5.17)

Let R: IR^{n} → IR^{n} be the identity transformation if ν = n, and, if ν < n, an orthogonal linear transformation such that

R(Σ(C)) = IR^{ν} × {0_{n−ν}}  (5.18)

0_{n−ν} being the origin of IR^{n−ν}. For every u ∈ K_{C}^{1,∞} we define the functions u′ and û as

u′(y) = u(R^{−1}y) y ∈ IR^{n},  (5.19)

û(y_{1}, …, y_{ν}) = \{ \begin{matrix}u(y_{1}, …, y_{n}) & if ν = n \\ & & (y_{1}, …, y_{ν}) ∈ IR^{ν}. \\ u′(y_{1}, …, y_{ν}, 0_{n−ν}) & if ν < n \end{matrix}  (5.20)

Since R^{−1} = R^{T} we have that

D_{y}u′(y) = D_{x}u(R^{−1}y)R^{−1} = D_{x}u(R^{−1}y)R^{T} = (RD_{x}u(R^{−1}y)^{T})^{T} for a.e. y ∈ IR^{n},

(5.21)

that is

D_{y}u′(y) ∈ IR^{ν} × {0_{n−ν}} for a.e. y ∈ IR^{n},  (5.22)

hence by (5.22) and (5.18) it follows that u′ effectively depends only on (y_{1}, … y_{ν}) when (y_{1}, …, y_{n}) varies in IR^{n} and that

Dû(y_{1}, …, y_{ν}) ∈ Pr_{ν}(RC) for every (y_{1}, …, y_{ν}) ∈ IR^{ν},  (5.23)

Pr_{ν} being the projection function from IR^{n} to IR^{ν} defined by Pr_{ν}(y_{1}, …, y_{n}) = (y_{1}, …, y_{ν}) for every (y_{1}, …, y_{n}) ∈ IR^{n}.

In conclusion by (5.17) and (5.18) we can also assume that 0_{ν}, the origin of IR^{ν}, belongs to the interior (in IR^{ν}) of Pr_{ν}(RC), i.e.

0_{ν} ∈ (Pr_{ν}(RC))^{∘}.  (5.24)

We have

Lemma 5.3. Let K be a subset of D′ verifying (5.1) (ii), (5.2), (5.3) and (5.6); let C be defined by (5.5) and K_{C}^{1,∞} by (3.2) with p = +∞. Then for every u ∈ K_{C}^{1,∞}

û ∈ C_{0}^{∞}(IR^{ν}) ⇒ u ∈ K.  (5.25)

Proof. Let u be in K_{C}^{1,∞} and t in ]0,1[. Let R be the above defined mapping, then by virtue of (5.2) and Proposition 5.1, Pr_{ν}(RC) turns out to be convex; moreover, being Du(x) ∈ C for a.e. x in IR^{n}, by (5.24) we infer

tDû(y_{1}, …, y_{ν}) ∈ (Pr_{ν}(RC))^{∘} for every (y_{1}, …, y_{ν}) ∈ IR^{ν}.  (5.26)

By (5.23) and Proposition 2.8 applied with m = ν and G = Pr_{ν}(RC), using the convention that dist(z, ∅) = +∞ for every z ∈ IR^{m}, we get that there exists δ > 0 such that

dist(tDû(y_{1}, …, y_{ν}), ∂ Pr_{ν}(RC)) > δ for every (y_{1}, …, y_{ν}) ∈ IR^{ν}.  (5.27)

A. Corbo Esposito, R. De Arcangelis / A characterization of sets of functions ... 185

Since û ∈ C_{0}^{∞}(IR^{ν}), let {û_{h}}_{h} be a sequence of piecewise affine functions on IR^{ν} such that

\{ \begin{matrix}û_{h} ∈ L^{∞}(IR^{ν}) & for every h ∈ IN, \\ û_{h} → tû & uniformly on IR^{ν} \\ Dû_{h} → tDû & in (L^{∞}(IR^{ν}))^{ν} \end{matrix}  (5.28)

(see for example Proposition 2.1 at page 309 in [34]), then by (5.26), (5.27) and (5.28) it follows that for h large enough

Dû_{h}(y_{1}, …, y_{ν}) ∈ Pr_{ν}(RC) for a.e. (y_{1}, …, y_{ν}) ∈ IR^{ν}.  (5.29)

By (5.28) if we define the functions u′_{h} and u_{h} by

\{ \begin{matrix}u′_{h}(y_{1}, …, y_{n}) = û_{h}(y_{1}, …, y_{ν}) \\ u_{h}(y) = u′_{h}(Ry), y ∈ IR^{n} \end{matrix}  (5.30)

we deduce that {u_{h}}_{h} is a sequence of piecewise affine functions on IR^{n} such that

u_{h} → tu in W^{1,∞}(IR^{n}),  (5.31)

moreover by (5.29) we have that for h large enough

Du_{h}(x) ∈ C for a.e. x ∈ IR^{n}.  (5.32)

By (5.32) and (5.16) we deduce that for h large enough

u_{h} ∈ K  (5.33)

hence by (5.33), (5.31) and (5.6) we get

tu ∈ K for every t ∈]0, 1[.  (5.34)

As t ↦ 1^{−} by (5.34) and (5.6) the thesis follows. By Lemma 5.3 we deduce the following result. Lemma 5.4. Let K be a subset of D′ verifying (5.1) (ii), (5.2), (5.3), and (5.6); let C be defined by (5.5) and K_{C}^{1,∞} by (3.2) with p = +∞. Then for every u ∈ K_{C}^{1,∞}

û ∈ C^{∞}(IR^{ν}) ∩ L^{∞}(IR^{ν}) ⇒ u ∈ K.  (5.35)

Proof. Let u be in K_{C}^{1,∞} and t in ]0,1[. As in the proof of the Lemma 5.3 there exists δ > 0 such that (5.27) holds. For every h ∈ IN let φ_{h} ∈ C_{0}^{∞}(IR^{ν}) be such that

\{ \begin{matrix}0 ≤ φ_{h} ≤ 1, φ_{h}(x) = 1, for every x such that |x| ≤ h \\ ∥Dφ_{h}∥_{L^{∞}(IR^{ν})^{ν}} ≤ \frac{δ}{∥û∥_{L^{∞}(IR^{ν})} + 1} \end{matrix}  (5.36)

and define the functions ŵ_{h} as

ŵ_{h}(y_{1}, …, y_{ν}) = φ_{h}(y_{1}, … y_{ν})û(y_{1}, …, y_{ν}) (y_{1}, …, y_{ν}) ∈ IR^{ν}.  (5.37)

By (5.36) we soon have

tDŵ_{h}(y_{1}, … y_{ν}) ∈ Pr_{ν}(RC) for every (y_{1}, …, y_{ν}) ∈ IR^{ν},  (5.38)

hence, if we define the functions w′_{h} and w_{h} in the same way of (5.30), by (5.38) and Lemma 5.3 we deduce that

tw_{h} ∈ K for every h ∈ IN.  (5.39)

By (5.36) and (5.37) we soon have that

w_{h} → u in W_{loc}^{1,∞}(IR^{n}),  (5.40)

hence by (5.40), (5.6) and (5.39) we obtain that

tu ∈ K for every t ∈]0, 1[.  (5.41)

As t ↦ 1^{−} by (5.41) the thesis follows.

Lemma 5.5. Let K be a subset of D′ verifying (5.1) (ii), (5.2), (5.3) and (5.8); let C be defined by (5.5) and K_{C}^{1,∞} by (3.2) with p = +∞. Then

K_{C}^{1,∞} ∩ L^{∞}(IR^{n}) ⊆ K.  (5.42)

Proof. Let u ∈ K_{C}^{1,∞} ∩ L^{∞}(IR^{n}), ϵ > 0 and let u_{ϵ} be the regularization of u given by (2.7). Since Du(x) ∈ C for a.e. x in IR^{n}, by b) of Proposition 5.1 and Proposition 2.7 it turns out that Du_{ϵ}(x) ∈ C for every x in IR^{n}; therefore we have that

u_{ϵ} ∈ K_{C}^{1,∞} ∩ C^{∞}(IR^{n}) ∩ L^{∞}(IR^{n}).  (5.43)

By (5.43) it soon follows that û_{ϵ} ∈ C^{∞}(IR^{ν}) ∩ L^{∞}(IR^{ν}), hence by (5.43) and Lemma 5.4 we deduce that

u_{ϵ} ∈ K for every ϵ > 0.  (5.44)

By (5.44) and (5.8) it follows that u ∈ K, that is the thesis.

We can now prove the characterization result in the case of Lipschitz continuous functions.

Theorem 5.6. Let K be a subset of D′ verifying (5.1)÷(5.3), (5.8); let C be defined by (5.5) and K_{C}^{1,∞} by (3.2) with p = +∞. Then

K ∩ W_{loc}^{1,∞} = K_{C}^{1,∞}.  (5.45)

Proof. By virtue of Proposition 5.2 we only have to prove that

K_{C}^{1,∞} ⊆ K ∩ W_{loc}^{1,∞}.  (5.46)

To this aim take u in K_{C}^{1,∞}. For every h ∈ IN we define the functions u_{h} as

u_{h}(x) = max{−h, min{u(x), h}} x in IR^{n},  (5.47)

then we soon have

Du_{h}(x) = Du(x) if |u_{h}(x)| ≤ h, Du_{h}(x) = 0 if |u_{h}(x)| > h.  (5.48)

By (5.48) and (5.17) we have

u_{h} ∈ K_{C}^{1,∞} ∩ L^{∞}(IR^{n}) for every h ∈ IN,  (5.49)

hence by (5.49) and Lemma 5.5 we deduce

u_{h} ∈ K for every h ∈ IN.  (5.50)

By (5.50) and (5.8) inclusion (5.46) and the thesis follows as h → +∞.

6. The characterization result in the general case

In the present section we complete the results of the previous one by characterizing the set K when it is contained in a topological vector subspace U of D′. We assume that (U, τ) is a Hausdorff locally convex topological vector subspace of D′ such that

u ∈ U, c ∈ IR, y ∈ IR^{n}, t > 0 ⇒ (i) u + c ∈ U, (ii) T [y]u ∈ U, (iii) O_{t}u ∈ U; (6.1)

\{ \begin{matrix}(i) & τ is less fine than w^{∗} − W_{loc}^{1,∞}(IR^{n}) on U ∩ W_{loc}^{1,∞}, \\ (ii) & τ is finer than w^{∗} − D′(IR^{n}); \end{matrix}  (6.2)

for every u ∈ U the function y ∈ IR^{n} ↦ T [y]u ∈ U is continuous.  (6.3)

We first need to deduce some properties of the regularizations of the elements of U.

Lemma 6.1. Let (U, τ) be a sequentially complete Hausdorff locally convex topological vector subspace of D′ verifying (6.1) (ii), (6.2) (ii) and (6.3). Let α be a mollifier in (2.6), assume that spt(α) is convex and let, for every u ∈ U, ϵ > 0, u_{ϵ} be defined by (2.6). Then

u_{ϵ} ∈ (U, τ)_{seq} − cl(conv({T [y]u: y ∈ B_{ϵ}})) for every u ∈ U, ϵ > 0;  (6.4)

u_{ϵ} → u in τ as ϵ → 0^{+} for every u ∈ U.  (6.5)

Proof. Let us first prove (6.4). Let u ∈ U and ϵ > 0, then by (6.3) it follows that the function y ∈ IR^{n} ↦ α(y)T [ϵy]u ∈ U is continuous with compact support hence, by Proposition 2.4, it is τ-integrable on IR^{n}. By (6.2) (ii) it follows that the above function is also w^{∗}−D′(IR^{n}) integrable on IR^{n} hence, by Proposition 2.5, the distributions u_{ϵ} and ∫_{IR}^{n} α(y)T [ϵy]u dy agree; by virtue of this, (6.1) (ii) and Proposition 2.6 we deduce (6.4).

Let us prove now (6.5). Let u ∈ U, let {p_{a}}_{a∈A} be a family of seminorms on U generating the topology τ and let B be a bounded open set containing the support of α.

By using (6.4) for every a ∈ A the assumptions of Proposition 2.3 are satisfied with E = B, f = α(·)T [ϵ·]u and Φ = p_{a}, hence by Proposition 2.3 we deduce

p_{a} (∫_{IR_{n}} α(y)T [ϵy]u dy − u) = p_{a} (∫_{B} α(y)(T [ϵy]u − u)dy) ≤
≤ ∫_{B} α(y)p_{a} (T [ϵy]u − u) dy for every a ∈ A.

(6.6)

By (6.3) we get that

sup_{y∈B} p_{a}(T [ϵy]u − u) → 0 as ϵ → 0^{+} for every a ∈ A,  (6.7)

hence by (6.6) and (6.7) we obtain

∫_{IR_{n}} α(y)T [ϵy]u dy → u in τ as ϵ → 0^{+}.  (6.8)

At this point we observe that by Proposition 2.5 the distributions u_{ϵ} and ∫_{IR}^{n} α(y)T [ϵy]u dy agree, hence (6.5) follows by (6.8) and Proposition 2.5.

About the set K we assume that K is contained in U and that

K is τ-sequentially closed.  (6.9)

Lemma 6.2. Let (U, τ) be a sequentially complete Hausdorff locally convex topological vector subspace of D′ verifying (6.1) (ii), (6.2) (ii), (6.3) and let K be a subset of U satisfying (5.1) (ii), (5.2) and (6.9). Then

K = (U, τ)_{seq} − cl(K ∩ W_{loc}^{1,∞}).  (6.10)

Proof. Since K ∩ W_{loc}^{1,∞} ⊆ K, by (6.9) we soon deduce that

(U, τ)_{seq} − cl(K ∩ W_{loc}^{1,∞}) ⊆ K.  (6.11)

We now prove the reverse inclusion in (6.11). Let u ∈ K, ϵ > 0 and y ∈ B_{ϵ}.

By (5.1) (ii) it follows that

T [y]u ∈ K,  (6.12)

hence by (6.12) and (5.2) we get

conv({T [y]u: y ∈ B_{ϵ}}) ⊆ K,  (6.13)

therefore by (6.13), (6.1) (ii) and (6.9) we conclude that

(U, τ)_{seq} − cl(conv({T [y]u: y ∈ B_{ϵ}})) ⊆ K.  (6.14)

By (6.4) of Lemma 6.1 and (6.14) we obtain, provided that the mollifier in u_{ϵ} has convex support,

u_{ϵ} ∈ K ∩ W_{loc}^{1,∞},  (6.15)

hence by (6.15) and (6.5) of Lemma 6.1 we get as ϵ → 0^{+}

K ⊆ (U, τ)_{seq} − cl(K ∩ W_{loc}^{1,∞}).  (6.16)

By (6.16) and (6.11) equality (6.10) follows.

We now prove the main result of this paper. Let us recall that given a subset C of IR^{n}, the set K_{C} is defined by

K_{C} = {u ∈ D′: −⟨u, Dφ⟩ ∈ C for every φ ∈ D with φ ≥ 0, ∫_{IR_{n}} φ = 1} .  (6.17)

Theorem 6.3. Let (U, τ) be a sequentially complete Hausdorff locally convex topological vector subspace of D′ verifying (6.1)÷(6.3). Let K be a subset of U satisfying (5.1)÷(5.3) and (6.9). Let C be given by (5.5) and K_{C} by (6.17). Then C is closed, convex and

K = K_{C} ∩ U.  (6.18)

On the contrary, given a closed convex subset C of IR^{n} and defined the set K_{C} by (6.17), it turns out that conditions (5.1)÷(5.3) and (6.9) are satisifed by K = K_{C} ∩ U.

Proof. By (6.9) and (6.2) (i) condition (5.6) holds, hence by Proposition 5.1 C turns out to be closed and convex. Let us now observe that by (6.2) (i) condition (5.8) holds, hence by Theorem 5.6 we deduce that

K ∩ W_{loc}^{1,∞} = K_{C}^{1,∞} = K_{C}^{1,∞} ∩ U,  (6.19)

K_{C}^{1,∞} being given by (3.2) with p = +∞. At this point, by (6.2) (ii) and Lemma 6.1, the assumptions of Proposition 4.1 are fulfilled, hence by Lemma 6.2, (6.19) and Proposition 4.1 equality (6.18) follows. Finally the last part of the thesis follows by (6.1), (6.2) (ii) and Proposition 3.5.

190 A. Corbo Esposito, R. De Arcangelis / A characterization of sets of functions ...

7. Applications to some spaces

In the present section we apply Theorem 6.3 when U agrees with the most common spaces in Analysis. Theorem 7.1. Let K be a subset of D′, respectively of S′, verifying (5.1)÷(5.3) and (6.9) with (U, τ) equal to (D′, w^{∗} − D′(IR^{n})), respectively to (S′, w^{∗} − S′(IR^{n})). Let C be defined by (5.5), K_{C} by (6.17) and K_{C}^{S} by (3.7). Then C is closed, convex and

K = K_{C},  (7.1)

respectively

K = K_{C}^{S}.  (7.2)

On the contrary, given a closed convex subset C of IR^{n} and defined the set K_{C} by (6.17), respectively K_{C}^{S} by (3.7), it turns out that conditions (5.1)÷(5.3) and (6.9) with (U, τ) equal to (D′, w^{∗} − D′(IR^{n})), respectively to (S′, w^{∗} − S′(IR^{n})), are satisfied by K = K_{C}, respectively by K = K_{C}^{S}.

Proof. The thesis follows by Theorem 6.3 applied with (U, τ) equal to (D′, w^{∗}−D′(IR^{n})), respectively to (S′, w^{∗} − S′(IR^{n})) and by Proposition 3.4, once observed that the above function spaces verify conditions (6.1)÷(6.3) and are sequentially complete.

Theorem 7.2. Let p ∈ [1, +∞] and let K be a subset of L_{loc}^{p} verifying (5.1)÷(5.3) and (6.9) with (U, τ) equal to (L_{loc}^{p}, L_{loc}^{p}(IR^{n})) if p ∈ [1, +∞[ and to (L_{loc}^{∞}, w^{∗} − L_{loc}^{∞}(IR^{n})) if p = +∞. Let C be defined by (5.5), K_{C} by (6.17) and K_{C}^{p} by (3.5).

Then C is closed, convex and

K = K_{C}^{p}.  (7.3)

On the contrary, given a closed convex subset C of IR^{n} and defined the sets K_{C}^{p} by (3.5), it turns out that conditions (5.1)÷(5.3) and (6.9) with (U, τ) equal to (L_{loc}^{p}, L_{loc}^{p}(IR^{n})) if p ∈ [1, +∞[ and to (L_{loc}^{∞}, w^{∗} − L_{loc}^{∞}(IR^{n})) if p = +∞ are satisfied by K = K_{C}^{p}.

Proof. The thesis follows by Theorem 6.3 applied with (U, τ) equal to (L_{loc}^{p}, L_{loc}^{p}(IR^{n})) if p ∈ [1, +∞[ and to (L_{loc}^{∞}, w^{∗} −L_{loc}^{∞}(IR^{n})) if p = +∞ and by Proposition 3.3, once observed that the above spaces verify conditions (6.1)÷(6.3) and are sequentially complete.

Theorem 7.3. Let K be a subset of C^{0}, respectively of M_{loc}, verifying (5.1)÷(5.3) and (6.9) with (U, τ) equal to (C^{0}, C_{loc}^{0}(IR^{n})), respectively to (M_{loc}, w^{∗} − M_{loc}(IR^{n})). Let C be defined by (5.5), K_{C}^{0} by (3.4) and K_{C}^{M} by (3.6). Then C is closed, convex and

K = K_{C}^{0},  (7.4)

respectively

K = K_{C}^{M}.  (7.5)

On the contrary, given a closed convex subset C of IR^{n} and defined the set K_{C}^{0} by (3.4) and the set K_{C}^{M} by (3.6), it turns out that conditions (5.1)÷(5.3) and (6.9) with (U, τ) equal to (C^{0}, C_{loc}^{0}(IR^{n})), respectively to (M_{loc}, w^{∗} − M_{loc}(IR^{n})), are satisfied by K = K_{C}^{0}, respectively by K = K_{C}^{M}.

Proof. The thesis follows by Theorem 6.3 applied with (U, τ) equal to (C^{0}, C_{loc}^{0}(IR^{n})), respectively to (M_{loc}, w^{∗} − M_{loc}(IR^{n})) and by Proposition 3.3, once observed that the above spaces verify conditions (6.1)÷(6.3) and are sequentially complete.

Theorem 7.4. Let K be a subset of BV_{loc} verifying (5.1)÷(5.3) and (6.9) with (U, τ) equal to (BV_{loc}, w^{∗} − BV_{loc}(IR^{n})). Let C be defined by (5.5) and K_{C}^{BV} by (3.3).

Then C is closed, convex and

K = K_{C}^{BV} .  (7.6)

On the contrary, given a closed convex subset C of IR^{n} and defined the set K_{C}^{BV} by (3.3), it turns out that conditions (5.1)÷(5.3) and (6.9) with (U, τ) equal to (BV_{loc}, w^{∗}−BV_{loc}(IR^{n})) are satisfied by K = K_{C}^{BV}.

Proof. The thesis follows by Theorem 6.3 applied with (U, τ) equal to (BV_{loc}, w^{∗} − BV_{loc}(IR^{n})) and by Proposition 3.2, once observed that this space verifies conditions (6.1)÷(6.3) and is sequentially complete.

Theorem 7.5. Let p ∈ [1, +∞] and let K be a subset of W_{loc}^{1,p} verifying (5.1)÷(5.3) and (6.9) with (U, τ) equal to (W_{loc}^{1,p}, W_{loc}^{1,p}(IR^{n})) if p ∈ [1, +∞[ and to (W_{loc}^{1,∞}, w^{∗} −W_{loc}^{1,∞}(IR^{n})) if p = +∞. Let C be defined by (5.5) and K_{C}^{1,p} by (3.2).

Then C is closed, convex and

K = K_{C}^{1,p}.  (7.7)

On the contrary, given a closed convex subset C of IR^{n} and defined the sets K_{C}^{1,p} by (3.2), it turns out that conditions (5.1)÷(5.3) and (6.9) with (U, τ) equal to (W_{loc}^{1,p}, W_{loc}^{1,p}(IR^{n})) if p ∈ [1, +∞[ and to (W_{loc}^{1,∞}, w^{∗} − W_{loc}^{1,∞}(IR^{n})) if p = +∞ are satisfied by K = K_{C}^{p}.

Proof. Let us observe that if p ∈ [1, +∞[, by (5.2), the set K is W_{loc}^{1,p}(IR^{n}) closed if and only if it is w − W_{loc}^{1,p}(IR^{n}) closed, hence the thesis follows by Theorem 6.3 applied with (U, τ) equal to (W_{loc}^{1,p}, w − W_{loc}^{1,p}(IR^{n}) if p ∈ [1, +∞[ and to (W_{loc}^{1,∞}, w^{∗} − W_{loc}^{1,∞}(IR^{n})) if p = +∞ and by Proposition 3.1, once observed that the above spaces verify conditions (6.1)÷(6.3) and are sequentially complete.

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